Question 6.15: A FET with gate periphery W = 300 μm has the following intri......
A FET with gate periphery W = 300 μm has the following intrinsic parameters: C_{G S}=0.1 \times 10^{-12} \text{F} , C_{G D} \approx 0, C_{D S}=0.002 \times 10^{-12} \text{F} , R_I+R_G=5 \; \Omega, \text{ g}_m=0.02\text{ S} , R_{D S}=556 \; \Omega . Moreover, L_D=50 \times 10^{-12} \text{H} , L_G=100 \times 10^{-12} \text{H}. Derive the parameters per unit length of the device, exploited to design an ideal, continuous distributed amplifier. Evaluate the additional drain capacitance needed to achieve synchronous coupling between the drain and gate line, and the frequency behavior of the voltage amplification as a function of the device length (periphery) L.
The FET specific parameters are:
\begin{aligned} \mathcal{C} _G & \approx \mathcal{C} _{G S}=C_{G S} / W=3.33 \times 10^{-10}\text{ F/m} \\ \mathcal{C} _{G D} & \approx 0. \\ \mathcal{C} _{D S} & =C_{D S} / W=6.67 \times 10^{-12}\text{ F/m} \\ \mathcal{R} _G & =\left(R_I+R_G\right) \;W=0.0015 \;\; \Omega \cdot\text{ m} \\ \mathcal{G} _m & =\text{g}_m / W=66.67\text{ S/m} \\ \mathcal{R} _D & \approx \mathcal{R} _{D S}=R_{D S} W=0.167 \;\; \Omega \cdot\text{ m} \\ \mathcal{L} _D & =L_D / W=1.67 \times 10^{-7}\text{ H/m} \\ \mathcal{L} _G & =L_G / W=3.33 \times 10^{-7}\text{ H/m}. \end{aligned}
To obtain synchronous coupling we should add to the drain a p.u.l. capacitance \mathcal{C} _a such that:
\mathcal{L} _G \mathcal{C} _{G S}= \mathcal{L} _D\left( \mathcal{C} _a+ \mathcal{C} _{D S}\right) \rightarrow \mathcal{C} _a=6.57 \times 10^{-10}\text{ F/m},
leading to a total drain p.u.l. capacitance:
\mathcal{C} _D= \mathcal{C} _a+ \mathcal{C} _{D S}=6.64 \times 10^{-10}\text{ F/m}.
We then have:
\tau_G=4.995 \times 10^{-13}\text{ s} , \quad \tau_D=1.1076 \times 10^{-10}\text{ s},
with related cutoff frequencies f_D=1.44\text{ GHz} , f_G=318\text{ GHz}; therefore in the microwave range the lossy LC approximation holds for the characteristic impedances of the gate and drain line:
Z_{0 G}=\sqrt{\frac{ \mathcal{L} _G}{ \mathcal{C} _G}}=22.394 \; \Omega, \quad Z_{0 D}=\sqrt{\frac{ \mathcal{L} _D}{ \mathcal{C} _D}}=15.859 \; \Omega
and for the losses:
Since the dominant attenuation is strongly frequency dependent, so is the optimum length. This parameter is therefore hardly useful in this case, since if a wideband response is intended the length should be of the order of the optimum one close to the maximum design frequency. For this reason it is more convenient to plot the amplifier response for different amplifier lengths as a function of frequency from (6.99),
see Fig. 6.54. We see that losses imply a decrease of the gain at high frequency. However, taking into account that the original cutoff frequency of the device was f_T=32\text{ GHz}, we have with L = 5 mm a low-frequency gain of 8 dB with a 3 dB bandwidth of around 40 GHz, with a GBP around 100 GHz; with L = 10 mm a low-frequency gain of 14.5 dB with a 3 dB bandwidth of around 27 GHz, with a GBP around 140 GHz. We see therefore an effective increase of the GPB with the conventional device. As a last remark, notice that the gain drop at high frequency is dominated by the \omega^2 behavior of the gate line losses.
